Question

Sum of the digits in the value of $\left(1+\tan 5^{\circ }\right)\left(1+\tan 10^{\circ }\right)\left(1+\tan 15^{\circ }\right)\dots$ $\left(1+\tan 45^{\circ }\right)=$ is______.

Answer 1

Answer: 5

$(1+\tan \theta )\left(1+\tan \left(45^{\circ }-\theta \right)\right)$
$=(1+\tan \theta )\left(1+\frac{1-\tan \theta }{1+\tan \theta }\right)$
$=(1+\tan \theta )\left(\frac{2}{1+\tan \theta }\right)=2$
Hence
$LHS=2\left(1+\tan 5^{\circ }\right)\left(1+\tan 40^{\circ }\right)\left(1+\tan 10^{\circ }\right)\left(1+\tan 35^{\circ }\right)$
$\left(1+\tan 15^{\circ }\right)\left(1+\tan 30^{\circ }\right)\left(1+\tan 20^{\circ }\right)\left(1+\tan 25^{\circ }\right)$
$=2\cdot 2^4=2^5$